There are three great reasons to watch live sports. First, a live sporting event is the ultimate reality show: unpredictable and entertaining. Second, watching athletes at the top of their game is inspiring. Third, sports is math come to life.

  • Batting average of .333 – baseball
  • Field goal percentage of 56.2% – basketball
  • Goals against average of 1.65 – hockey
  • Greens in regulation – 79.86 – golf

These are but four examples of the math that pervades major sports. To understand the decisions and strategies made in the heat of a live game, one must understand the mathematics behind the game. What are the chances of making the 3-pointer? Should the pitcher walk the .333 batter with runners in scoring position?

Of course, you can’t watch sports without hearing the ridiculous statistics as well. For me, it is a bit of a game to decide which of the bizarre statistics drummed up during the game have any meaning. Baseball, with ample dead air space to fill, has the most outlandish – and least useful – stats. (Football is a close second.)

“He is the first pitcher in 24 years to win 70 percent of his decisions while logging more strikeouts than innings pitched.” (More crazy stats:

On the other hand, some stats can be just plain bizarre: Two consecutive baseball games with no RBIs (runs batted in) for either team did actually happen…

As a girl growing up watching sports, it did not matter to me whether the players were male or female – I was still interested. For today’s youngsters, however, girls can look up to stars like Christine Sinclair (soccer), Hayley Wickenheiser (hockey) and Eugenie Bouchard (tennis). Bouchard’s inspired play in the Australian Open won many fans around the world.  Something to chew on: Bouchard’s service game was her downfall in the semi-final – Bouchard: 45 per cent first serves in and 18 per cent second serves won verus Li: 61 per cent first serves in and 41 per cent second serves won.

Try watching sports and discussing the stats with your children – it might even get you as hooked on sports (& math) as I am…But just for the record, playing sports beats watching, hands down!!

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Hippocratic Oath for Math Teachers

Parallel to the Hippocratic oath, which is broadly interpreted as “Do no harm,” there should be one for math teachers –“Do not instill fear and loathing of math.” Alas, there is no such oath for aspiring teachers, and we are all seeing the results.

Ontario students, whose performance in math has been on the decline according to the province’s own EQAO (Education Quality and Accountability Office) standards over the past 5 years, are now lagging behind internationally as well, according to the latest study of math and science by the Organization for Economic Cooperation and Development (OECD).

Who is shocked that students in Quebec, who learn from the best-trained math teachers, fare better than students in all other provinces in international rankings? Not coincidentally, this is also the province where discovery learning is shunned in favour of  more “old-fashioned” methods.

Some writers, including Gwyn Morgan in the Business section of the Globe and Mail on December 16, 2013, see two options for math: discovery learning versus rote learning.

I believe strongly that there is a happy medium – I shall call it Understanding and Practice. Math learning should be built on understanding the concept, then practice to become proficient, as measured by accuracy and speed.

What does this look like? Using multiplication as an example, kids should learn via skip counting: for the 7s, kids would count, then recite or write: 7, 14, 21, 28, 35, etc. Once they understand this, they practice – often. After considerable practice, they would be tested to see if they can answer all questions correctly. Once accuracy is achieved, students would work on speed. This is in sharp contrast to the rote approach of memorizing that 7 x 5 = 35, 9 x 4 = 36, etc. and the discovery approach that they just need to know that the answer involves multiplication and that they could figure it out in time.

For many students, homework is the greatest problem in their math education. Teachers routinely assign homework plagued with one of the following obstacles:

  1. Material is not taught in class
  2. No example is given to show how to complete it
  3. Answers are not taken up

For some unfortunate students, they routinely face all 3 obstacles…

Here is what I propose:

Teachers should assign homework that builds:

  1. Confidence – by assigning work that has been taught
  2. Accuracy – by showing them how to complete it
  3. Speed – by giving multiple questions involving the same processes

Parents should question any homework that does not satisfy these 3 criteria.

Not only do students who are lacking in confidence not progress with math, they develop fear and loathing towards this important subject. Math learning is cumulative  – and confidence is the foundation for future learning. This is particularly true for any students with learning disabilities or students who are lagging behind – building confidence is the crucial first step towards learning.

Accuracy is under-rated in the discovery approach. Students I tutor are routinely struggling with basic math operations (addition, subtraction, multiplication and division), which is holding them back from developing more complex math skills. At a more basic level, many students are struggling with subtraction because they haven’t fully learned addition. The parallel problem is that students are struggling to learn division because they haven’t understood and mastered multiplication.

Speed of calculation is not simply a measure of how well the student can complete the work at hand, it is strongly linked to future success in more complex math areas. For example, a student who can calculate the multiplication fact quickly can free up his or her working memory to work on the rest of the equation (A student struggling with 6 x 8 or 9 x 7 will not have the time on a math test to complete this question!)

(6x + 9y)(8x + 7y)=

As usual, parents are left with the dilemma of how to help their children with math, and many enlist tutors. As much as I enjoy being a math tutor, I deplore the need for my services. I would much rather spend time teaching teachers…perhaps I will try this in 2014…I need a new challenge!

Teri Courchene

December 16, 2013

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Fixing Education in Ontario

Ok – so I am breaking my own rules about spreading out my blog posts over time. Everyone who knows me knows that I am terrible at following rules. I blame my Montessori education for that…

Here is a report I submitted today as part of the Ontario Ministry of Education’s initiative to improve education:

Of course, we were supposed to answer specific questions, but true to form, I proposed my own ideas instead:

Three Initiatives for Ontario Schools

  1. Self-regulation is key in the early years
  2. Early years math – focus on numeracy
  3. Narrow the fundraising gaps among schools


Parents and educators looking for the key to the future academic performance of children should examine self-regulation. In contrast to early reading performance, which has not been shown to be related to later academic achievement, self-regulation is integrally linked to future success, not only in academics but also in health and prosperity (Diamond & Lee, 2011; Moffitt et al, 2011). Teaching and assessment should be heavily weighted towards fostering strong self-regulation rather than on academic achievement in the kindergarten years.

Recommendation: Evaluate self-regulation in kindergarten and intervene to boost these skills if they are lacking


To paraphrase George Orwell, “all math strands are created equal, but some are more equal than others.”

Why is early-years math important? Math is a cumulative subject, in which prior learning is the foundation for future learning. Gaps that emerge in achievement in the early years tend to grow over time.  This has been dubbed “the Matthew effect” – strong students get stronger and weaker students lag increasingly far behind. On school report cards in Ontario, no priority is given to any one of the five math strands.

The gaps in achievement between the highest and lowest-achieving students are larger in the United States than in Canada. Over the past ten years, several major U.S. policy reports have recommended that among the five major strands: number sense and numeration, measurement, patterning and algebra, data management and probability, geometry and spatial sense, number sense and numeration should be the primary focus (NCTM, 2006; NRC, 2009). Simple game-playing with a mathematically accurate game board can generate strong gains in math achievement in a very short time (Siegler & Ramani, 2009).

Recommendation: Focus on number sense and numeration in kindergarten and in the primary years


Although in comparison to the United States, Canada has much less income inequality and less inequality in academic achievement between students of high-income versus low-income status, Canadian educators should not be complacent. Faring better than the abysmal performance of U.S. schools is hardly cause for celebration.

One area in which there is a legislative gap that can be closed by the provincial Ministry of Education is in fundraising by parents. It is widely known that parent-led fundraising in some high-income primary schools in Toronto generates between $50,000 and $100,000 in additional revenues for schools with between 300-500 students. These additional funds are used to purchase furniture (chairs, lunch room tables), programs (drama, arts, science), technology (smart boards for each classroom), and fund safety programs (safe arrival).  Although there are some rules surrounding how these funds may be used and whether the school or the school board owns the items purchased with these funds, not all of these rules are routinely followed.

There is an opportunity for a province-wide initiative to recognize that parents can and should raise money for their child’s school but, at the same time, share the wealth with less-affluent schools. It could be mandated that 25 per cent of all funds raised be provided to a fund for less-affluent Home and School Associations/Parent-Teacher Associations to enable them to provide similar opportunities in less-affluent neighbourhoods. Although there could be an initial backlash against this, it should be remembered that this fundraising is done not in a private school, but in a public school, on property owned by the school board. The strongest argument for revenue sharing comes back to the students – altruism is a core part of each student’s education.

Recommendation: Introduce rules forcing revenue sharing among parent-led school associations.


Teri Courchene

BA (UWO), MA (Queen’s), AMI Montessori Diploma (FME), MEd (OISE)

Toronto, ON





Diamond, A., & Lee, K. (2011). Interventions shown to aid executive function development in children 4 to 12 years old. Science, 333, 959-964.

Moffitt, T.E., Arsenault, L., Belsky, D., Dickson, N., Hancox, R.J., Harrington, H., Houts, R., Poulton, R., Roberts, B. W., Ross, S., Sears, M. R., Murry Thomson, W., & Caspi, A. (2011). A gradient of childhood self-control predicts health, wealth, and public safety. Proceedings of the National Academic of Science (PNAS), 108, 2693-2698.

National Council of Teachers of Mathematics (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence, Reston, VA: NCTM.

National Research Council (2009). Mathematics Learning in Early Childhood: Paths toward excellence and equity, Committee on Early Childhood Mathematics. C.T. Cross, T.A. Woods and H. Schweingruber (Eds.).Washington, DC: The National Academies Press.

Siegler, R.S., & Ramani, G.B. (2009). Playing linear board games – but not circular ones – improves low-income preschoolers’ numerical understanding. Journal of Educational Psychology. Vol.101, No. 3, 545-560.

Statistics Canada (2009). Canadian Nine-year-olds at School. Special Surveys Division, September 25, Statistics Canada Catalogue no. 89-599-M, no. 6. Retrieved from: MIE2009006&lang=eng#formatdisp


Other research papers on these topics:




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It’s a mystery

It is a mystery to my family why I keep making things for fundraisers, for students, and for fellow teachers…but it is not a mystery to me. These days so many items that surround us are electronic and mass-produced and it is a pleasure to have hand-made items crafted with natural materials. But why do I make so many things? Easy to answer…it makes me very happy to give them away! In some cases I have to sell them (just to be able to replenish my reserves of fabric, thread and yarn…).

This week, I delivered 18 mystery bags to the students at the Foundation for Montessori Education. This institution is where I completed my training as a Montessori Teacher in 2004. The school is AMI accredited (the true Montessori brand) and it is a top-notch program that makes the students work themselves to the bone for 9 months learning about children and how they develop and learn. In all of my years and 3 university degrees, I have never come close to working as hard as I did that year. Truth be told, I had also never been inspired to work so hard or enjoyed any endeavour so much. This year’s students are a lovely group and it was a treat to spend a bit of time with them. Here are the choices they made for fabrics for their various mystery bags….


Are you wondering what these bags are for? In a Montessori classroom, the mystery bag contains items that the children are familiar with. The bag is taken to a table, and the child puts his/her hand in, grasps an object and names it before removing the object from the bag. All of the items are lined up from left to right across the table. It is one of the most amazing sights to see the delight on the child’s face when completing this activity.


This year, I have very young students – ages 2-3 1/2 – and we have not tried this activity yet. Today, however, with my newly-made mystery bag, I think I will show the eldest one. She is ready, and I know I am looking forward to it…


P.S. Today I receive my third university degree, my Masters of Education from OISE. It ranks second in learning and enjoyment, behind my AMI Montessori diploma, and miles ahead of my BA and MA in economics!


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Lightest Pumpkin Cheesecake Ever

This recipe has been tested and tweaked three times and the consensus is – this one is just right. It is a cross between a pumpkin pie (spicy and not too sweet) and a cheesecake (graham crust and tangy taste of cream cheese). The egg white on the graham crust keeps it slightly crunchy and the walnut and honey topping adds a decadent touch to this very light dessert.

Image 1

Graham crust:

1 cup graham cracker crumbs

1/4 cup packed brown sugar

3 tbsp melted margarine

1 egg white, beaten

1. In a medium bowl, prepare crust:

  • Combine crumbs, sugar and melted margarine and mix well.
  • Press into an 9-inch ungreased springform pan, pressing crust down firmly with the back of a spoon.
  • Bake at 325 degrees for 10 minutes.
  • Remove from oven and brush with about half of the egg white.
  • Return to the oven and bake for 5 more minutes.


1 pkg light cream cheese (250g)

3 eggs, one at a time

1/2 cup granulated sugar

1 1/2 cups canned pumpkin

1/2 tsp cinnamon

1/2 tsp nutmeg

1/2 tsp ginger

1/8 tsp salt

1/2 cup light (0-2%) sour cream

2. In a large bowl, prepare filling:

  • With electric mixer, beat cream cheese until light and fluffy.
  • Add eggs one at a time, beating well.
  • Add sugar and beat well.
  • Add remaining ingredients, mixing well.
  • Pour filling into cooked pie crust, spreading filling evenly.

3. Bake pie:

  • Bake at 325 degrees for 35 minutes.
  • Turn the oven off and leave the cheesecake in for 10 minutes.
  • Cheesecake is done when the edges look cooked and the centre jiggles slightly.

4. Topping: optional

Nutmeg, 1/4 cup liquid honey, 1/2 cup chopped walnuts

  • Sprinkle cheesecake lightly with nutmeg
  • Chop walnuts and roast on ungreased pan at 350 degrees for 5-10 minutes.
  • Heat honey in microwave, then stir in roasted walnuts.
  • Spread onto cooked cheesecake, making sure to cover any cracks…



  • Pie is best prepared ahead and chilled.
  • Much lighter than a traditional cheesecake.
  • Lightly spiced and not too sweet. For a sweeter cheesecake, increase sugar to 2/3 cup.

Hope you try it and let me know if you like it…


Here is a pdf of the recipe:

Lightest Pumpkin Cheesecake Ever

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Making counting more fun

As usually happens, whenever I have an assignment due at OISE, it feels like a good time to make another post. Perhaps because I am drowning in papers and ideas I don’t fully understand (yet), it is satisfying to post about something I actually know something about!

Over the past two years, I have been making and using counting strings in the classroom with children ages 2-6. Even though Montessori classrooms have an abundance of hands-on mathematics materials, there was something lacking for the young children learning to count. Counting is more than reciting the string of numbers. Children need to have a concept of quantity and need to be able to make the one-to-one correspondence between the number they say and the object they are counting.


To bridge this gap, I borrowed a tool from the golf world – a stroke counter. Thankfully my golf came has improved since my teenage years, when a stroke counter with 10 beads would have come in quite handy on virtually every hole!


Why do we need an aid for children learning to count? English has irregular patterns in numbers for the teens, and for 20, 30, and 50.  Unlike in many Asian languages, which would use ten-three for 13, and two-tens for twenty, we have to learn new words. Practice with these numbers is required to build fluency and accuracy in counting. The reason the strings I make go up to 50 is that after 50, all of the number patterns are regular – sixty, seventy, eighty and ninety are all predictable.


The number strings I make for the children are simple, and embody the following characteristics:

  • beads are attached to the string and are slid along as they are counted
  • each decade (set of ten beads) is of one colour
  • colours adjacent to each other are contrasting
  • string for very young children has 10 beads
  • string for children ages 2-3 has 20 beads
  • string for children ages 3-6 has 50 beads

When the children are in class with me, I get them to choose the colours of the beads we use and count out 10 of each. When the string is completed, I show the child how to count along the string, then have the child count and slide each bead along as he/she counts. I make a note about the child’s progress and/or which numbers require prompts. That day, the child takes the counting string home to keep. Quite often, I make a 20-bead string and the child counts so well that we progress to a 50-bead string very quickly.

So, how do you make one?

To make a 20-bead string:

  • Select 10 beads each of 2 contrasting colours.
  • Cut a piece of string about 3 feet long.
  • Fold the string in half and tie a knot 2 inches from the fold.
  • Use a bit of glue on each end of the string to make the end stiff.
  • Slide each end into opposite sides of one bead, then slide the bead along to the knot.
  • Add the next bead in the same way, sliding it down next to the previous bead.
  • When all the beads are added, make a knot about 1 1/2 inches away from the final bead.
  • Trim the ends of the string to about 1/2 to 1 inch away from the final knot.

IMG_0668I have shared these strings and ideas with Kindergarten teachers – and they have found them useful in the classroom.  Let me know if you find that they work.

Teri Courchene

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Tutoring works!

This year has been the first time I have tried weekly tutoring with kids in addition to teaching classes of young children.

Why is tutoring so rewarding?

  • I can build a strong cooperative relationship with the child.
  • Our time together is a respite – a quiet time in a supportive environment.
  • I can target the work to meet exactly what the child needs – whether it is blending sounds for an early reader or understanding the relationship between fractions and percentages – using hands-on materials wherever possible.
  • As the child develops proficiency at his/her work, his/her confidence grows.
  • With greater confidence, we can venture into new areas of learning.
  • Every once in a while, I take a minute and show the child the things that he/she can do well – too many people have usually told the child what he/she does not do well…
  • Don’t tell the older kids, but the younger ones get some time to choose what they want to do in the classroom…making it more interesting.
  • So far, the children and I are enjoying ourselves…and both of us are learning…which is why tutoring is becoming my main line of work.

Teri Courchene


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